Current flowing through a wire decreases linearly from 10A to zero in 4s as shown in the graph. Find the total charge flowing through the wire in the given time interval.
Show Hint
For any linear change in current from \( I_{\text{initial}} \) to \( I_{\text{final}} \), the total charge can also be calculated using average current:
\( Q = I_{\text{avg}} \times \Delta t = \frac{I_{\text{initial}} + I_{\text{final}}}{2} \times \Delta t \).
In this specific case, \( Q = \frac{10 + 0}{2} \times 4 = 5 \times 4 = 20 \text{ C} \).
Step 1: Understanding the Question:
The total charge flowing through a circuit component is equal to the area under its current-time (\( I-t \)) graph. Step 2: Key Formula or Approach:
The charge \( Q \) is given by the definite integral of current over time:
\[ Q = \int I(t) dt \]
Geometrically, this corresponds to the area of the region between the graph and the time axis. Step 3: Detailed Explanation:
The provided graph forms a right-angled triangle with the following dimensions:
- Base (\( \Delta t \)) = 4 s
- Height (initial current \( I_0 \)) = 10 A
The area of a triangle is calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values:
\[ Q = \frac{1}{2} \times 4 \text{ s} \times 10 \text{ A} = 20 \text{ C} \] Step 4: Final Answer:
The total charge flowing through the wire is 20C.
